This invention was funded by the U.S. Army Research Office and the Advanced Research Projects Agency. The government may have certain rights in this invention.
The present invention relates to an optical system which generates light beams having special properties at the output of an optical system. More specifically, the optical system of the present invention uses a special kind of radial harmonic function, preferably formed on a Fourier hologram, to produce a non-diverging beam in its far field.
Optical systems often have many important and diverse applications. Lasers, for example, allow a beam of light to be projected to a precise location, still maintaining a high intensity.
It is a common inaccurate belief that a laser emits a collimated or non-diffracting beam. In actuality, while the beam is more collimated than light from an ordinary lamp, in fact the laser beam is a Gaussian beam. The Gaussian beam diverges according to the rule that the spread angles are related to the wavelength of the light divided by the minimal lateral width of the beam. The Rayleigh distance xcex94z, or non-diffracting beam width of this Gaussian is expressed as:       Δ    ⁢          xe2x80x83        ⁢    z    ∝            W      2        λ  
where w is the initial beam width and xcex is the wavelength. This equation shows that reducing the initial width of the beam actually increases the spreading of the beam.
One aspect of the present invention avoids these phenomena by producing a beam which has better non-diffracting characteristics than a laser beam. The resulting beam has pseudo-non-diffracting properties, which is to say that the beam does not diffract over a defined interval. A pseudo non-diffracting beam (xe2x80x9cPNDBxe2x80x9d) behaves as a non-diffracting beam only in the sense that, in contrast to a Gaussian beam, its peak intensity remains almost constant along an axial intervalxe2x80x94its xe2x80x9clengthxe2x80x9d. The beam full width in half maximum (xe2x80x9cFWHMxe2x80x9d) increases much more slowly than expected from an ordinary Gaussian beam.
This is coupled with another problem which the inventors recognized as existing in the art. When a plane wave crosses a simple spherical lens, that plane wave is focused to the lens"" front focus. The distance between the front focus and the lens is called the focal length. The focal point is on the optical axis of the lens, which is usually an axis that is parallel to the light rays, and that passes through the center of the lens.
A simple lens produces an output which has a bell-curve-shaped intensity profile along the optical axis. Such a system would show a strong intensity precisely at the focal point. That intensity falls off rapidly on both sides of the focal point. A lens can be used to focus the information onto the focal point. However, even a slight deviation from the distance of the focal point will cause results which vary greatly from that desired. In a typical optical system, this results in a sharp image at the focal point, and a blurred image off the focal point.
The width of the bell curve-shaped intensity profile at the output of a lens is sometimes called the depth of focus of the lens. Some lens systems allow wider depths of field, making it easier to stay within the focal point of the lens. Various techniques are known for increasing the depth of focus.
The output characteristics of an optical system are defined according to the so-called point spread function. This point spread function tells us about the output of the system when there is a point at the input. The information is calculated by convolving the input function i with the function h. For a normal point spread function, the output is sharp and clear in the imaging plane.
It is one objective of the present invention to change the ordinary behavior of light when it passes through a lens and optical system to form a pseudo-non-diffracting beamxe2x80x94essentially a very wide depth of focus. The present invention defines an optical system that has a filter in the path of the light prior to the lens. The overall optical system forms a pseudo-non-diffracting beam over a specified position. Put another way, the present invention allows obtaining a very long point spread function.
The term xe2x80x9csword beamxe2x80x9d as used in this specification refers to a beam that maintains a constant intensity along a defined optical axis for an arbitrary interval. The lateral width at any cross section of the sword beam is substantially the same as that of the focused beam from an ordinary lens.
It is known that the depth of field of an optical system can be increased by placing an aperture with a narrow ring at the rear focal plane of the lens. See, for example, T. Durnin, JOSA A 4651 (1987). FIG. 1 shows a schematic of that system. That rear focal plane is located at a focal distance from the lens, but on the same side of the lens from which the light arrives.
This system, however, has a number of drawbacks. Only a very small portion of the light passes through the narrow ring and becomes useful at the front focus. Most of the energy of the light is absorbed by the opaque regions of the mask. This results in an extremely inefficient optical system.
Others have suggested preprocessing the optical energy using a Fresnel plate. However, each section of the Fresnel plate has a different frequency. The plate is not radially symmetric, causing certain problems in the output.
For instance, when such a plate is used as a spatial filter of an imaging system, the quality of the output image will be different from one radial direction to another. That is because the output image is obtained from a convolution between the input image and a non-symmetric point spread function (the Fourier transform of the spatial filter distribution). Another example is producing a long narrow tunnel in some material by using a beam emerged from such a plate. The lateral cross section of the tunnel will not be radially symmetric if the beam is not radially symmetric.
Another suggestion has used an alternate technique to provide a desired output. The iterative technique defines constraints of the input and output domains, expresses the distances between the two by an error function, and iteratively finds a hologram that minimizes the error under the constraints. The iterative technique is described, for example, in J. Rosen; A. Yariv; xe2x80x9cSynthesis of an arbitrary axial field profile by Computer-Generated Hologramsxe2x80x9d, Optics Letters, Vol 11, #19, pp. #843-845.
Certain of these suggestions have been made by the present inventors, so no admission is made herein that this iterative technique is, in fact, prior art.
It is an object of the present invention to provide a totally new solution to these problems, using a completely new technique. A preferred technique of the present invention uses a special mask at the rear focal plane of the lens. This mask has a light-altering function formed thereon. This function is preferably a radial harmonic function, and more preferably a phase-only function, e.g., a radial harmonic function of the 4th order or greater. The optical system forms a desired pseudo non-diffracting beam having desired characteristics at a specified position. That position can be the front focal point of the lens or other positions. The properties of the sword beam can be controlled using various parameters of the filter.
The inventors have found that a radial harmonic function of the fourth order produces substantially improved effects since virtually no light is absorbed by the mask. A 4th order harmonic system operates as a phase-only filter to change the wave""s phase distribution when it passes through the filter. Other techniques, including other orders of radial harmonic function, however, can also be used. More generally, any approximate solution to the ideal Bessel function can be used, and preferably a solution made using the known mathematical theory of stationary phase approximation.
The method of stationary phase approximation is a mathematical technique to solve (approximately) integrals of the form ∫ab f(t)exp[jkxcexc(t)]dt. The method is based on the fact that for a large k the significant contribution to the integral result comes from the area under the function f(t)exp[jku(t)] around the stationary points. A stationary point to is the point where the derivative of xcexc(t) is zero. For example, if xcexcxe2x80x2(to)=0, but xcexcxe2x80x3(to) xe2x89xa00 and to is the only stationary point inside the interval [a,b], the integral is approximately equal to             ∫      a      b        ⁢                  f        ⁢                  (          t          )                    ⁢              exp        ⁡                  [                      j            ⁢                          xe2x80x83                        ⁢            k            ⁢                          xe2x80x83                        ⁢                          μ              ⁡                              (                t                )                                              ]                    ⁢              ⅆ        t              ≈            f      ⁡              (                  t          0                )              ⁢          exp      ⁡              [                  j          ⁢                      xe2x80x83                    ⁢          k          ⁢                      xe2x80x83                    ⁢                      μ            ⁡                          (                              t                0                            )                                      ]              ⁢                            2          ⁢          π                          k          ⁢                      xe2x80x83                    ⁢                                    μ              xe2x80x3                        ⁢                          (                              t                0                            )                                            ⁢                  exp        ⁡                  (                      j            ⁢                          xe2x80x83                        ⁢                          π              /              4                                )                    .      
More details about this method are described in: A. Papoulis, Systems and transforms with applications in optics, Ch. 7, 222-254, (McGraw-Hill 1968).
The present invention also describes techniques which change the original radial harmonic function to shift the sword beam within the optical space, even tilting it by up to 20xc2x0 from the optical axis and/or changing the position relative to the focal point. These shifting techniques allow the focal line for the light to be modified in any desired way.
Another aspect of the invention makes use of the shifting technique to allow the focal line to be a line of any desired shape. The inventors have titled this arbitrary-shaped focal line a xe2x80x9csnake beamxe2x80x9d.
The sword beam has important applications in imaging systems and/or in any application where a laser is or can be used. The output of an imaging system is an observed object which is located at the input plane. Its image is obtained at the image plane. If the image is observed at a distance away from the image plane, it looks out of focusxe2x80x94it looks like a blurred image of the original object.
The radial harmonic filter system of the present invention keeps the image pattern in focus for a longer distancexe2x80x94a distance equal to the length of the pseudo non-diffracting beam. Since this beam is shaped like a sword, it is sometimes called a sword beam. This system effectively increases the focal length of the imaging system. This technique has many applications as described herein.
A well-known telescopic system can be used according to the present invention. This system uses two lenses, one behind the other, with a common focal plane. When the image is placed in the rear focal plane of the first lens, the image plane becomes the front focal plane of the second lens. The radial harmonic filter is displayed between the two lenses at their common focal plane. When the two image systems are compared, the image in the system with the filter is clear and sharp while the other one is not.